A repeated rule can settle, alternate or grow
What determines whether errors around a fixed point shrink?
Entering midway? A recurrence defines the next state from the current state. A fixed point is unchanged by the update.
Start with a concrete question
Iterate x(next)=a×x+b for up to twelve exact discrete-model steps; this is not yet an approximation to continuous time.
Try the initial inputs, then write a prediction before changing one setting. All calculations run locally on your CPU.
Go directly to the experiment · Review prerequisite mathematics and return here
What do you expect?
How will behavior differ for a=0.5, a=−0.5 and a=1.5?
Change one thing. Inspect what happens.
- At defaults compute the first four values and solve for the fixed point.
- Use a negative multiplier of magnitude below one and inspect alternating errors.
- Try a=1 and distinguish b=0 from nonzero b.
- Try magnitude above one and compare starting at a fixed point with a small perturbation.
Your settingsMechanism
Calculation belowEvidence
Compare outcomes
Exact discrete recurrence on a fixed scale
Horizontal: Step number. Vertical: State.
- 1. State sequence (solid line)
Axes use the displayed ranges.
Read plotted values as a table
| Series | Step number | State |
|---|---|---|
| State sequence | 0 | 0 |
| State sequence | 1 | 1 |
| State sequence | 2 | 1.5 |
| State sequence | 3 | 1.75 |
| State sequence | 4 | 1.875 |
| State sequence | 5 | 1.938 |
| State sequence | 6 | 1.969 |
| State sequence | 7 | 1.984 |
| State sequence | 8 | 1.992 |
Follow the calculation
- Step 1: 0.5×(0.00000)+1=1.00000.
- Step 2: 0.5×(1.00000)+1=1.50000.
- Step 3: 0.5×(1.50000)+1=1.75000.
- Step 4: 0.5×(1.75000)+1=1.87500.
- Step 5: 0.5×(1.87500)+1=1.93750.
- Step 6: 0.5×(1.93750)+1=1.96875.
- Step 7: 0.5×(1.96875)+1=1.98438.
- Step 8: 0.5×(1.98438)+1=1.99219.
- Fixed point solves x=ax+b: x*=b/(1−a)=2.00000. Error obeys e(next)=a×e(now).
Errors shrink toward the fixed point; a negative multiplier alternates their sign.
Captures keep your inputs, prediction and observation. Results are recalculated from those inputs. Resetting inputs keeps your captures and challenge history.
Explain the mechanism
Start with the meaning
A recurrence defines the next state from the current state. A fixed point is unchanged by the update.
Iterate x(next)=a×x+b for up to twelve exact discrete-model steps; this is not yet an approximation to continuous time.
A worked explanation
A fixed point xsatisfies x=ax+b, so x=b/(1−a) when a≠1.
At a=0.5,b=1, the fixed point is 2. Starting at zero gives 1,1.5,1.75,1.875. Each error halves.
Subtracting the fixed-point equation shows e(next)=a e. Magnitude below one contracts errors; negative a alternates their sign. Exactly starting at an unstable fixed point can conceal sensitivity to perturbations.
Symbols and relationship
xₙ₊₁=axₙ+b; x*=b/(1−a) for a≠1; eₙ₊₁=aeₙ. For a=1,b=0 every state is fixed; for a=1,b≠0 none is.
Use the graph and trace together
The graph and calculations use your current inputs. Read axis labels and units before comparing positions or slopes. Change one input, predict the effect, and use the numerical trace to verify the plotted behavior. A displayed curve is an illustration; the formula and its assumptions determine which conclusions are justified.
Scope and edge cases
- Linear discrete recurrence, bounded horizon and inputs; arithmetic uses binary64.
- Fixed broad axes accommodate unstable cases and may make small stable changes visually subtle; read the trace as well.
Try a fresh case
For x(next)=0.75x+2, what is the fixed point?
This is a public practice check. Retries support learning; matching a number does not certify independent proficiency.
Take the investigation further
Choose any extension. These are project prompts for you to carry out; they are not automatically assessed.
School investigation
Create a table for positive and negative multipliers.
College investigation
Prove the closed-form solution and analyze boundary multipliers ±1.
Workplace investigation
Inspect a repeated adjustment policy for fixed points and perturbation amplification.
Keep the evidence, choose your next step
Completion records your own decision that you have worked through this investigation. Practice checks and independent project evidence are separate.